Self-Induced Cyclic Reorganization of Many Bodies Through Thermal Convection
Bin Liu1 and Jun Zhang1,2∗
1
Center for Soft Matter Research, Department of Physics,
New York University, 4 Washington Place, New York, NY 10003, USA
2
Applied Mathematics Laboratory, Courant Institute,
New York University, 251 Mercer Street, New York, NY 10012, USA
We investigate the dynamics of a thermally convecting fluid as it interacts with freely moving
solid objects. This is a previously unexplored paradigm of many-body interactions mediated by
thermal convection, which gives rise to surprising robust oscillations between different large-scale
circulations. Once begun, this process repeats cyclically, with the collection of spheres entrained
and packed from one side of the convection cell to the other. The frequency of the cycle is highest
when the spheres occupy about half of the cell bottom and their size coincides with the thickness
of the thermal boundary layer. This phenomenon shows that a deformable mass can stimulate a
turbulent, thermally convecting fluid into oscillation, a collective behavior that may be found in
nature.
PACS numbers: 47.27.te, 47.20.Bp, 05.65.+b
A fluid, when heated from below and cooled at the
top, is subject to buoyancy instability; subsequently,
Rayleigh-Bénard convection arises as the warmer fluid
ascends and the cooler fluid descends. A large-scale circulation of the bulk fluid emerges from the turbulent flow
as the system exceeds a finite threshold of external forcing [1–7]. The forcing intensity is characterized by the
dimensionless Rayleigh number [8], Ra = αg∆T H 3 /νκ,
where g is the acceleration due to gravity, ∆T is the temperature difference between bottom and top, H is the
depth of the fluid, and α, κ and ν are the thermal expansion coefficient, the thermal diffusivity, and the kinematic viscosity of the fluid, respectively. The threshold
Rayleigh number for a large-scale flow to appear is on
the order of 107 [2, 9]. At the top and the bottom plates,
thin thermal boundary layers exist, within which convection is suppressed by the no-slip condition at the fixed
boundaries. Within these regions, thermal conduction
alone provides the vertical heat transfer. The thickness
of the thermal boundary layer λth is observed to have different scaling dependencies on the Rayleigh and Prandtl
numbers ( Pr = ν/κ ) in various regimes, as summarized
by Grossmann and Lohse [10, 11]. In the regime of our
experiment, Ra ∼ 108−9 and 1 < Pr < 102 , the scaling
relationship is given by the measurement of Ashkenazi
and Steinberg [12], λth /H = 2.3Ra−0.3 Pr0.2 .
Turbulent thermal convection has been one of the most
studied physical dynamical processes in recent decades
[13, 14]. Its study has revealed fundamental and generic
features of systems driven from equilibrium and into
chaos [15, 16], and many open questions still remain, especially on the heat transport at high Rayleigh numbers
[11, 17, 18] and on the dynamical stability of the largescale circulation [19, 20]. However, most experimental
and theoretical studies of thermal convection have dealt
with fixed boundary conditions: all the solid walls are
rigid, and they do not respond to or interact with the
FIG. 1: The experimental setup. A thermal convection cell
(A), shown in gray, is surrounded by another box that contains the same fluid and subject to the same cooling (C) and
heating (B), so that the inner convection cell is well insulated at all vertical walls. Nylon spheres (D) are able to roll
freely on the smooth, mirror-quality bottom as they are entrained by the convecting fluid. The system we study consists of the inner convection cell and the enclosed spheres. A
pair of vertically separated thermistors (E) measures the bulk
temperature, and flow velocity through the cross-correlation
between the two signals. Conformal transformation of the
images taken by a video camera (F), produces top-viewed
spheres patterns.
fluid flows confined within. Previous experiments[21–24],
meant to simulate a continent drifting over mantle convection, have shown coupled motion of a single floating
boundary interacting with the fluid underneath. The experiment discussed here aims to investigate the interac-
2
tion between a collection of many bodies – on the order
of a few hundreds – and the surrounding convective fluid.
As shown in Fig. 1, a Rayleigh-Bénard convection cell
of 20 × 18.4 × 7.6 cm ( H × L × W ), containing a waterglycerol mixture (ρ = 1.115 g/ml), is heated from below
by a DC electric heater and cooled at its top with a water
circulator. The relatively smaller width (W ) of the cell
allows only two possible large-scale circulations to exist
at high Rayleigh numbers (Ra > 107 ): a clockwise circulation or counter-clockwise one (from the perspective of
the camera) that occupies the entire cell. To eliminate
lateral heat exchange through the side walls, this convection cell is contained within a bigger box which holds the
same fluid and subject to the same cooling and heating.
The freely moving objects introduced into the inner
convection cell are nylon spheres, which are a few millimeters in diameter (d) and 2% denser (ρ′ = 1.134±0.002
g/cm3 ) than the fluid. A large number of the spheres –
N , on the order of a few hundreds – form a single loose
layer at the bottom. They can move along the mirrorquality bottom as each sphere experiences fluid drag from
the convective flow. The typical Stokesian drag force,
Fdrag ∼ 6πηdU , is on the order of a few hundredth of a
dyne, given the sphere size, the speed of the convecting
flow, U , and its viscosity, η.
In the absence of the spheres, either the clockwise or
the counter-clockwise large-scale circulation is stable up
to a time-scale of around 40 hours. Beyond this timescale, the circulation can flip direction spontaneously in
a random fashion. This is consistent with observations
made earlier in a cylindrical cell as the relative rare cessation events took place [19, 20], and such flow reversals
are regarded as a result of hydrodynamical fluctuations
overcoming a potential barrier that separates the possible
states [25].
Having added the spheres to the convection cell, however, we find a drastic acceleration of these flow reversal
events. Driven by the reversing circulation, the spheres
are packed and unpacked cyclically. Figure 2 shows a
sequence of sphere positions during this process. Under
large-scale circulation, the spheres are entrained by the
flow and packed at one end of the cell bottom (Figs. 2a
and 2c). This aggregation is made progressively, often
one sphere at a time or with small clusters, by the flow
that travels across the bottom. Typical packing patterns
include thin strips of square lattice aligned with the sidewalls and small patches of triangular patterns near the
interior of the pack. These patches are pieced together
in a random fashion as they evolve in time.
As the flow reverses direction, the spheres become unpacked (Figs. 2b, 2d). Figures 3a and 3b show that
unpacking of the aggregate starts soon after the circulation reverses direction. Shown in Fig. 2e, flow reversal
happens only after the majority of spheres have collected
on one side. Even though these flow reversals take place
with varying degree of randomness, depending on each
FIG. 2: Interaction between 426 spheres of diameter 3.2 mm
and the convective fluid at Ra = 3.2 × 109 . (a-d): four successive snapshots of the spheres during one cycle of two adjacent reversals. Two instances (a and c) show relatively dense
packing, and two others (b, d) show the transient period as
the aggregate is unpacked. Each of the snapshot is averaged
along the short side of the convection cell to obtain a onedimensional density profile. The evolution of such a profile
(e) demonstrates cyclic packing and unpacking of the mobile
spheres, while the four horizontal lines indicating the four
instances shown on the left.
experimental condition (Ra and N ), an average reversal
interval can be well-defined.
The migration of an individual sphere takes up to a few
tens of seconds, which is 2 - 5 times the duration needed
for the circulation to flow across the bottom. However, it
can take considerable time for a dense packing to assemble. Jamming of spheres during the packing phase often
yields vacancies (e.g.: the one seen near the right end of
Fig 2a). The extent and location of these vacancies fluctuate in time and they are likely to be filled before the
next reversal. It is conceivable that the randomness in
the process of achieving dense packing is the main source
of randomness in the timing of flow reversal.
The longest time-scale in the oscillation is apparently
that between flow reversals. Once the circulation changes
direction, it achieves its full strength in a much shorter
time-scale, typically within a few flow circulations (Fig.
3a). Due to the coupling between the thermally convective flows and the free-moving spheres, relatively stable
large-scale circulations are interrupted by abrupt flow reversal events and followed by the slow and accumulative
aggregation of free-moving bodies. Figure 3b shows the
correlation between the flow velocity and the position of
the aggregate as a whole.
The spheres play an important role in inducing the
circulation to reverse, and they collectively cause the
coupled system to oscillate. The frequency of the selfsustained oscillation depends on the effectiveness of the
perturbation introduced by the spheres. Figure 3c illus-
3
FIG. 3: Correlation between the large-scale circulation and
the collection of the spheres. Here, N = 510, d = 3.2 mm
and Ra = 1.8 × 109 . Graph (a) shows a time series of the
circulation velocity, switching in time between two circulatory states. Graph (b) indicates that the center of mass of all
spheres varies in phase with the clockwise circulation (white
bands) and the counter-clockwise circulation (grey bands).
Illustration (c) shows the interaction between the large-scale
circulation and the spheres. Once packed, the spheres collectively counteract the circulation by suppressing the local heat
flux (relatively strong heat flux shown as vertical arrow). The
large-scale flow eventually reverses its direction. The new circulation unpacks the spheres and drives them to the other
side of the convection cell, where they are reassembled. Once
initiated, this process repeats cyclically.
trates the feedback mechanism that is likely to operate
in our coupled solid-fluid system. In our earlier work,
we have shown that a solid boundary of similar material
and thickness reduces the local heat transfer by a factor of order 10, because it prohibits local convective heat
transfer [22, 24]. This effect is similar to the thermal
blanketing effect of a solid boundary studied in geophysical contexts [26, 27]. In the current experiment, the precise heat-transfer contrast between a sphere-packed area
and an unpacked area depends on the detailed packing
configuration. For a typical random packing, we estimate that such contrast is on the order of 3 - 5, based
on the volume occupied by the spheres compared with a
solid block. Collectively, the aggregate of spheres act as a
thermal blanket that reduces the local heat flux from the
bottom plate; it relatively enhances local heat influx at
the unoccupied areas. This contrast eventually overturns
the existing large-scale circulation and cyclic oscillations
thus emerge.
To modify the amplitude and nature of the perturbation to the system, we use different sizes of spheres, vary
the coverage ratio by adjusting the number of spheres,
and also vary the Rayleigh number. As we change the
boundary layer thickness by varying the Rayleigh number
while using the same group of spheres, we find that there
is a minimal flow reversal interval when λth ∼ d/2, as
shown in Fig. 4a. We conjecture that the thermal blan-
FIG. 4: The Rayleigh number and the number of free spheres
are varied. (a) The circulation reversal intervals as a function
of the Rayleigh number when d = 3.2 mm and N = 365. An
oscillation with shortest reversal interval is obtained when the
thickness of the thermal boundary layer, λth , is comparable
to the radius (d/2) of the spheres. (b) At Ra = 3.1 × 109 and
d = 3.2 mm, the dependence of the reversal interval on the
number of spheres. Nmax ( ∼ 840 ) is the number of spheres
needed to fill up the bottom with random packing [29]. The
inset shows the zoom-in detail of the data at N ∼ Nmax /2.
keting effect of an individual sphere is affected by its area
fraction at a height h from the bottom (the cross-section
area at height h normalized by its maximum cross-section
at height d/2), which is Γ = 4(d − h)h/d2 , where h < d.
When h = d/2, this expression takes its maximum value.
Our observation hence suggests that when the sphere radius (d/2) is close to λth , the activity at the edge of the
thermal boundary layer – the most active region in a
thermally convecting fluid [28] – is most efficiently suppressed.
We notice that the radius of the spheres (d/2 = 1.6
mm) is about 23% greater than the thickness of the thermal boundary layer (λth = 1.3 mm) when the maximum
reversal rate is observed [Fig. 4a]. In the following argument, we attempt to explain this mismatch and show
that the heat blocking effect is most effective when λth
is somewhat smaller than the sphere radius.
Nylon spheres have nearly the same heat diffusivity
as the surrounding fluid. For an individual sphere, the
vertical heat flux has to pass through both the solid
sphere (thickness of order d/2) and a thermal boundary layer (thickness of order λth ) that covers on the top
of the sphere. The contrast between the heat fluxes
with (j) and without (j0 ) the spheres can be characterized by number C = (j0 − j)/j0 = d/(2λth + d),
which ranges from 0 to 1 as for no heat flux blocking
effect to ideal blocking. The effective thermal blanketing
factor, Θ, is the combination (product) of the vertical
heat contrast C and the area fraction Γ taken by the
spheres, and
√ Θ = CΓ = 4λth (d − λth )/d(2λth + d). When
λth = d( 3 − 1)= 1.2mm, factor Θ takes its maximum
value and the heat blocking effect is most efficient. This
number is now more consistent with our experimental
observation.
The above simple argument has not taken into ac-
4
count of the complicated sphere geometry and the detailed packing patterns, which can only serve as a guide
to better understand the thermal blanketing effects introduced by the spheres.
Moreover, the number of spheres also affects the reversal rate; it is greatest when the spheres are able to cover
about half of the bottom plate (Fig. 4b). At this coverage, the spheres can create the highest contrast in heat
flux at the bottom. With either a smaller or greater number of spheres, the bottom becomes geometrically more
homogeneous and the heat contrast is reduced. Further,
with more spheres present in the system, the suppressed
mobility of spheres due to crowding also leads to longer
reversal interval.
We have shown that a self-sustained, reorganizing
state emerges as large-scale feedback (between many
bodies and the surrounding fluid) overrides chaotic
features intrinsic to a complex system. It is surprising
that the freely-moving bodies introduced here do not
destroy the existing coherent fluid structures, nor do
they elicit more complex patterns in the system, but
rather they provoke frequent switching between the two
circulatory states. These free bodies, to some degree,
are organized cyclically through passive response to
the fluid; indeed, the collection of many bodies in our
experiment behaves as a single deformable mass. We
expect yet richer dynamics to occur if a similar study
is conducted in cells with greater aspect ratios, where
several large-scale circulation regions and dynamical
states may co-exist.
We thank A. Libchaber, M. Shelley, S. Spagnolie, T.
Bringley, J. Percus and J.-Q. Zhong for helpful discussions. This work is supported in part by the Department
of Energy (Grant No. DE-FG0288ER25053).
Electronic address: [email protected]
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